3.321 \(\int \frac{(a+b x)^{9/2}}{x^5} \, dx\)

Optimal. Leaf size=116 \[ \frac{315}{64} b^4 \sqrt{a+b x}-\frac{315}{64} \sqrt{a} b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{105 b^3 (a+b x)^{3/2}}{64 x}-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{(a+b x)^{9/2}}{4 x^4}-\frac{3 b (a+b x)^{7/2}}{8 x^3} \]

[Out]

(315*b^4*Sqrt[a + b*x])/64 - (105*b^3*(a + b*x)^(3/2))/(64*x) - (21*b^2*(a + b*x
)^(5/2))/(32*x^2) - (3*b*(a + b*x)^(7/2))/(8*x^3) - (a + b*x)^(9/2)/(4*x^4) - (3
15*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64

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Rubi [A]  time = 0.111505, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{315}{64} b^4 \sqrt{a+b x}-\frac{315}{64} \sqrt{a} b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{105 b^3 (a+b x)^{3/2}}{64 x}-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{(a+b x)^{9/2}}{4 x^4}-\frac{3 b (a+b x)^{7/2}}{8 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^5,x]

[Out]

(315*b^4*Sqrt[a + b*x])/64 - (105*b^3*(a + b*x)^(3/2))/(64*x) - (21*b^2*(a + b*x
)^(5/2))/(32*x^2) - (3*b*(a + b*x)^(7/2))/(8*x^3) - (a + b*x)^(9/2)/(4*x^4) - (3
15*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64

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Rubi in Sympy [A]  time = 15.2699, size = 107, normalized size = 0.92 \[ - \frac{315 \sqrt{a} b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64} + \frac{315 b^{4} \sqrt{a + b x}}{64} - \frac{105 b^{3} \left (a + b x\right )^{\frac{3}{2}}}{64 x} - \frac{21 b^{2} \left (a + b x\right )^{\frac{5}{2}}}{32 x^{2}} - \frac{3 b \left (a + b x\right )^{\frac{7}{2}}}{8 x^{3}} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**5,x)

[Out]

-315*sqrt(a)*b**4*atanh(sqrt(a + b*x)/sqrt(a))/64 + 315*b**4*sqrt(a + b*x)/64 -
105*b**3*(a + b*x)**(3/2)/(64*x) - 21*b**2*(a + b*x)**(5/2)/(32*x**2) - 3*b*(a +
b*x)**(7/2)/(8*x**3) - (a + b*x)**(9/2)/(4*x**4)

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Mathematica [A]  time = 0.0800793, size = 86, normalized size = 0.74 \[ \frac{1}{64} \left (-\frac{\sqrt{a+b x} \left (16 a^4+88 a^3 b x+210 a^2 b^2 x^2+325 a b^3 x^3-128 b^4 x^4\right )}{x^4}-315 \sqrt{a} b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^5,x]

[Out]

(-((Sqrt[a + b*x]*(16*a^4 + 88*a^3*b*x + 210*a^2*b^2*x^2 + 325*a*b^3*x^3 - 128*b
^4*x^4))/x^4) - 315*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64

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Maple [A]  time = 0.021, size = 85, normalized size = 0.7 \[ 2\,{b}^{4} \left ( \sqrt{bx+a}+a \left ({\frac{1}{{x}^{4}{b}^{4}} \left ( -{\frac{325\, \left ( bx+a \right ) ^{7/2}}{128}}+{\frac{765\,a \left ( bx+a \right ) ^{5/2}}{128}}-{\frac{643\,{a}^{2} \left ( bx+a \right ) ^{3/2}}{128}}+{\frac{187\,\sqrt{bx+a}{a}^{3}}{128}} \right ) }-{\frac{315}{128\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^5,x)

[Out]

2*b^4*((b*x+a)^(1/2)+a*((-325/128*(b*x+a)^(7/2)+765/128*a*(b*x+a)^(5/2)-643/128*
a^2*(b*x+a)^(3/2)+187/128*(b*x+a)^(1/2)*a^3)/x^4/b^4-315/128*arctanh((b*x+a)^(1/
2)/a^(1/2))/a^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221874, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, \sqrt{a} b^{4} x^{4} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x + a}}{128 \, x^{4}}, -\frac{315 \, \sqrt{-a} b^{4} x^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x + a}}{64 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^5,x, algorithm="fricas")

[Out]

[1/128*(315*sqrt(a)*b^4*x^4*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(12
8*b^4*x^4 - 325*a*b^3*x^3 - 210*a^2*b^2*x^2 - 88*a^3*b*x - 16*a^4)*sqrt(b*x + a)
)/x^4, -1/64*(315*sqrt(-a)*b^4*x^4*arctan(sqrt(b*x + a)/sqrt(-a)) - (128*b^4*x^4
- 325*a*b^3*x^3 - 210*a^2*b^2*x^2 - 88*a^3*b*x - 16*a^4)*sqrt(b*x + a))/x^4]

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Sympy [A]  time = 33.399, size = 182, normalized size = 1.57 \[ - \frac{315 \sqrt{a} b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{64} - \frac{a^{5}}{4 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{13 a^{4} \sqrt{b}}{8 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{149 a^{3} b^{\frac{3}{2}}}{32 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{535 a^{2} b^{\frac{5}{2}}}{64 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{197 a b^{\frac{7}{2}}}{64 \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{9}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**5,x)

[Out]

-315*sqrt(a)*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/64 - a**5/(4*sqrt(b)*x**(9/2)
*sqrt(a/(b*x) + 1)) - 13*a**4*sqrt(b)/(8*x**(7/2)*sqrt(a/(b*x) + 1)) - 149*a**3*
b**(3/2)/(32*x**(5/2)*sqrt(a/(b*x) + 1)) - 535*a**2*b**(5/2)/(64*x**(3/2)*sqrt(a
/(b*x) + 1)) - 197*a*b**(7/2)/(64*sqrt(x)*sqrt(a/(b*x) + 1)) + 2*b**(9/2)*sqrt(x
)/sqrt(a/(b*x) + 1)

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GIAC/XCAS [A]  time = 0.213852, size = 149, normalized size = 1.28 \[ \frac{\frac{315 \, a b^{5} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 128 \, \sqrt{b x + a} b^{5} - \frac{325 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{5} - 765 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{5} + 643 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{5} - 187 \, \sqrt{b x + a} a^{4} b^{5}}{b^{4} x^{4}}}{64 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^5,x, algorithm="giac")

[Out]

1/64*(315*a*b^5*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 128*sqrt(b*x + a)*b^5
- (325*(b*x + a)^(7/2)*a*b^5 - 765*(b*x + a)^(5/2)*a^2*b^5 + 643*(b*x + a)^(3/2)
*a^3*b^5 - 187*sqrt(b*x + a)*a^4*b^5)/(b^4*x^4))/b